BA has for a long time made that word "Midrash" count for rather a lot. It's a useful way of sanctioning the exclusion of most of the Jewish-inclined sayings attributed to Jesus. Needless to say it reflects a deeply ingrained bias on the part of the scholars in question who adopt this approach.
It also goes without saying that a large number of distinguished scholars adopt an almost contrary approach, and emphasise the essential Jewishness of the historical Jesus. Schweitzer was very much in this camp, but the more modern proponents of the Jewish Jesus are figures such as Geza Vermes, E.P. Sanders, Bart Ehrman etc. (and just in case someone starts to pipe up; yes, I've read them )
There are quite a number of other takes on Jesus, as I'm sure you know: the Jesus Seminar approach, typified by John Dominic Crossan takes a very different view to the above. And there's also the 'Jesus the Magician'* approach. I'm sure all these differing scholars think they're being as objective as possible.
*The most notorious book adopting this approach was the work of Morton Smith, published in 1978 (this one I haven't read). There was a prolific poster on the old BBC site who was convinced that this take on Jesus was the correct one, and had written a university thesis on it, apparently.
With both Vermes and Sanders, there is at least superficial credibility about their views on Jesus' role as a very Jewish teacher and prophet, imbued, as they contend, in purely OT thinking. Their views are at face value, attractive; but the consensus seems to be that they are mistaken in regarding Jesus as a purely Jewish leader. There is one particular quote from Jesus, which neither of them suitably explains away:
And Jesus came and said to them, "All authority in heaven and on earth has been given to me. Go therefore and make disciples of
all nations, baptizing them in the name of the Father and of the Son and of the Holy Spirit, teaching them to observe all that I have commanded you. And behold, I am with you always, to the end of the age."
Also, if He was referring to Gentiles when he said, "I have other sheep that are not of this fold", then he must have already by that time put aside the idea that he "was sent only to the lost sheep of the House of Israel ," if He ever thought that.
One point worth mentioning, is that both these eminent Biblical scholars had no problem with the historicity, and indeed the divinity of Jesus. Perhaps some of the resident atheists here might benefit from a look at their work.